Number System

Natural Numbers:

The counting numbers 1,2,3----- are called natural numbers and is denoted by N.

Whole Numbers:

Whole numbers are all of the positive integers, including zero, which are not fractions or decimals. They are the numbers that are used to count and measure objects, such as 1, 2, 3, 4, 5, 6, etc.

Rational Numbers:

Rational numbers are any number that can be expressed as a fraction of two integers, with a non-zero denominator. Examples of rational numbers include fractions such as 3/4, -5/7, and 4/2.

A number of the form \( \frac{p}{q} \) , where q \( \neq\) 0, p and q are integers, and \( \frac{p}{q} \) is in the lowest form i.e., p and q have no common factors, is called a Rational Number.

Examples: \( \frac{3}{4} \), \( \frac{5}{7} \), \( \frac{4}{2} \) \( \frac{1}{2} \) etc.

Irrational number:

A non terminating and non recurring decimal is called irrational number.

An irrational number is a real number that cannot be expressed as a fraction or a ratio of two integers. Examples of irrational numbers include pi (π), the square root of two (√2), and the golden ratio (φ).

Examples: √2, √3, √5, √7, π, φ etc. /p>

Real Numbers:

The collection of rational numbers and irrational numbers is called set of real numbers.

Real numbers are numbers that can include fractions, irrational numbers, and integers. They can be represented on the number line and can be used to measure distances or to count. Real numbers can include both positive and negative numbers and can be rational or irrational.

Examples of real numbers include 0.5, 3/4, 0.00000031, -1.43, π, and √2.

Even Numbers:

A number which is divisible by 2 without any remainder is called an even number.

Examples of even numbers include 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, etc.

Odd Numbers:

A number which is not divisible by 2 is called an odd number.

Examples of odd numbers include 1, 3, 5, 7, 9, 11, 13, 15, and 17.

Prime Numbers:

A number, other than 1 is called a prime number, if it is divisible by 1 and itself only.

A prime number is a positive integer greater than 1 that is only divisible by 1 and itself.

Examples of prime numbers include 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, and 41.

Co-prime:

Co-prime, or relatively prime, numbers are two numbers that have no common factors other than 1.

For example, 4 and 7 are co-prime, but 4 and 6 are not co-prime because they both share the factor of 2.

For example, 8 and 15 are co-prime because their only common factor is 1, whereas 8 and 16 are not co-prime because they have a common factor of 2.

Examples of co-prime numbers are: 2 and 3, 5 and 7, 11 and 13, 17 and 19, 23 and 29, etc.

Twin-prime:

Twin primes are a pair of prime numbers that differ by two.

For example: , the numbers 3 and 5 are twin primes. Similarly, the numbers 17 and 19 are also twin primes.

Some other examples of twin primes are 5 and 7, 11 and 13, 29 and 31, 41 and 43, 59 and 61, 71 and 73, and 101 and 103.

Perfect number:

A perfect number is a positive integer that is equal to the sum of its proper positive divisors, excluding the number itself.
The smallest perfect number is 6, which is the sum of 1, 2 and 3, its proper divisors.

Example

The number 28 is a perfect number. Its proper divisors are 1, 2, 4, 7 and 14, and the sum of these is 28 (1 + 2 + 4 +7 + 14 = 28). Therefore, 28 is a perfect number.

Composite Numbers:

A number, other than 1, which is not prime, is called a composite number.

Composite numbers are any whole numbers greater than 1 that are not prime numbers.

Examples of composite numbers are 4, 6, 8, 9, 10, 12, 14, 15, and 16.

Important Facts:

1. All natural numbers are whole numbers.

2. All whole numbers are not natural numbers. 0 is a whole number which is not a natural number.

3. Even number + Even number = Even number

Odd number + Odd number = Even number

Even number + Odd number = Odd number

Even number - Even number = Even number

Odd number - Odd number = Even number

Even number - Odd number = Odd number

Odd number - Even number = Odd number

Even number x Even number = Even number

Odd number x Odd number = Odd number

Even number x Odd number = Even number

4. The smallest prime number is 2.

5. The only even prime number is 2. 6. The first odd prime number is 3.

7.1 is a unique number - neither prime nor composite.

8. The least composite number is 4.

9. The least odd composite number is 9.